DOCSLIB.ORG

Calculus Students' Understanding of Area and Volume Units

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Calculus Students' Understanding of Area and Volume Units INVESTIGATIONS IN MATHEMATICS LEARNING ©The Research Council on Mathematics Learning Fall Edition 2015, Volume 8, Number 1 Calculus Students’ Understanding of Area and Volume Units Allison Dorko Oregon State University [email protected] Natasha Speer University of Maine Abstract Units of measure are critical in many scientific fields. While instructors often note that students struggle with units, little research has been conducted about the nature and extent of these difficulties or why they exist. We investigated calculus students’ unit use in area and volume computations. Seventy-three percent of students gave incorrect units for at least one task. The most common error was the misappropriation of length units in area and volume computations. Analyses of interview data indicate that some students think that the unit of the computation should be the same as the unit specified in the task statement. Findings also suggest that some students have difficulties correctly indicating the units for computations that involve the quantity π. We discuss students’ correct and incorrect use of unit in relation to their understanding of area and volume as arrays, as well as in terms of Sherin’s Symbolic Forms. Keywords: undergraduate students, units, area, calculus Introduction Units of measurement are important in many disciplines including those in science, technology, engineering and mathematics (STEM). Values are often meaningless without units and there are often multiple options for the unit in which something is measured (e.g., meters, yards), creating potential ambiguities if quantities are reported without units. In addition, understanding - 23 - units can be useful for problem solving. For example, knowing what units the answer must be in can help one determine which quantities to combine to obtain the answer. In addition, as Rowland (2006) noted, “Units can also be used to check the validity of equations: while an equation can be wrong if the units are right, an equation can’t be right if the units are wrong” (p. 553). Many future scientists and engineers get their foundational understanding of key ideas of measuring quantities (e.g., rates, rates of change) in undergraduate calculus courses. Success in any STEM discipline requires both the knowledge to obtain quantitative answers and the knowledge of how those quantities relate to the physical world, including the units in which such quantities are measured. Findings from research on the teaching and learning of calculus indicate that applied problems as well as those involving modeling of quantities are quite challenging for students (e.g., Engelke, 2008; Martin, 2000; Orton, 1983). Despite the importance of understanding units of measure and anecdotal evidence from instructors that undergraduate students struggle with units, there is little research about the nature and extent of these difficulties. The few studies that exist provide evidence that units pose difficulties for undergraduate students in differential equations, physics, and chemistry (Redish, 1997; Rowland, 2006; Rowland & Jovanoski, 2004; Saitta, Gittings, & Geiger, 2011). Additionally, researchers have found that elementary school students have difficulty with measurement in general and the units of area and volume in particular (Battista & Clements, 1998a; Curry & Mitchelmore, 2006; Lehrer, 2003; Lehrer, Jacobson, Thoyre, Kemeny, Strom, Horvath, Gance, & Koehler, 1998; Lehrer, Jenkins, & Osana, 1998). The exploratory investigation reported here focused on calculus students’ understanding of the units of area and volume computations. Understanding what area and volume are and being able to carry out computations for them are important in the study of calculus. Topics such as the definition of the definite integral and the volumes of solids of rotation are built on ideas of area and volume. Understanding students’ thinking about units is one window into the understanding of these key issues and, to our knowledge, this is the first exploration of unit understanding for university calculus students. We were curious if the unit issues found in elementary school students persisted through their mathematical careers; given the findings about trouble with units in physics and differential equations, it seemed logical that they might. Knowing more about calculus students’ understanding of and difficulties with units of measurement can enhance the education field in two important ways. First, topics involving units are known to be difficult, but what is not clear is the extent to which the units aspect is causing the topics to be difficult. Second, armed with an understanding of the nature of those difficulties, instruction can be designed to help students develop more robust knowledge in this area in conjunction with learning the calculus content. - 24 - To begin an investigation of how calculus students understand units, we gave calculus students basic area and volume computational tasks with the goal of answering the following research questions: 1. How do calculus students answer area and volume computational tasks? 2. How do calculus students think about the units in area and volume computational problems? 3. How do calculus students’ difficulties with units compare to elementary school students’ difficulties with units? Analyses of written survey and interview data revealed that calculus students struggle with the units of area and volume computations; that these difficulties are related to students’ difficulties with understanding arrays and dimensionality; and that some calculus students’ difficulties are similar to those documented in elementary school students. Student Understanding of Units Investigating what students understand about units in mathematical contexts is of interest to science and mathematics educators alike. Units are an important part of many scientific calculations and findings from research indicate that students’ understanding of units affects their learning of some physics concepts (Rowland, 2006; Van Heuvelen & Maloney, 1999). While units may be less emphasized in mathematics, there are topics in which they play a critical role (e.g., mathematical modeling of physical phenomena, “word problems” of various sorts, measurement of one-, two-, and three- dimensional objects). In this section, we give examples of the importance of units in physics, chemistry, and mathematics, then review literature about elementary school students’ understanding of units. Units in Physics and Chemistry In disciplines with applied mathematics such as chemistry, physics, and engineering, the variables used in equations relate to physical quantities in the real world. Most of these quantities possess dimensions and these dimensions are an essential part of their nature. Density, for instance, is the amount of matter in a unit volume and thus thinking about density involves thinking about the units involved. Equations used for computation or modeling must both make mathematical sense and be dimensionally consistent. For instance, Newton’s Second Law (F=ma) is dimensionally consistent because the product of the units of mass and acceleration is kgm/s2, equivalent to Newtons (a unit of force). The Ideal Gas Law, PV = nRT, is an example of an equation in chemistry which must be dimensionally consistent. In chemistry, units are particularly important in stoichiometry and dimensional analysis. - 25 - Converting between units is especially problematic; in particular, “although unit conversions are taught in a variety of subjects over several grade levels, many students have not mastered this topic by the time they enter college” (Saitta, Gittings, & Geiger, 2011, p. 910). We did not locate any physics education articles explicitly stating that students struggle with units, but several researchers write that students do not make use of implying unitsʼ importance, difficulty, or both. Redish (1997) noted students may interpret symbols as pure numbers rather than as standing for physical quantities; students are also likely to substitute numbers into an equation right away. The implication is that students may miss opportunities to use the units of the quantity as a ‘clue’ in the equation (e.g., figuring out what other quantities might be needed). Van Heuvelen and Maloney (1999) proposed a way of using units to help physics students understand concepts. The researchers suggested that problems be written backwards; that is, rather than give students a physical situation and have them provide an equation to model it, the students are given an equation with quantities and their units and students create physical situations to match. They give the example of proposing to students the equation N-(60kg)(9.8 m/s2)=0 , from which students could reason that kg indicates mass and m/s2 indicates acceleration due to gravity, and so on. The students would then create scenarios, such as a 60kg person standing on a flat surface (Van Heuvelen & Maloney, 1999, p. 252-253). A strength of these problems is that “the units become more meaningful since they become the key to determining what quantities are involved… consequently the units become useful sources of information” (Van Heuvelen & Maloney, 1999, p. 254). Redish’s (2007) note that students, to their detriment, often try to work with only pure numbers and Van Heuvelen and Maloney’s (1997) emphasis on units as useful in decoding equations indicate the importance of units to the physics world, and their usefulness to students as conceptual tools. Units in Mathematics
Load more

Recommended publications
  • Measure, Integral and Probability
    Marek Capinski´ and Ekkehard Kopp Measure, Integral and Probability Springer-Verlag Berlin Heidelberg NewYork London Paris Tokyo Hong Kong Barcelona Budapest To our children; grandchildren: Piotr, Maciej, Jan, Anna; Luk asz Anna, Emily Preface The central concepts in this book are Lebesgue measure and the Lebesgue integral. Their role as standard fare in UK undergraduate mathematics courses is not wholly secure; yet they provide the principal model for the development of the abstract measure spaces which underpin modern probability theory, while the Lebesgue function spaces remain the main source of examples on which to test the methods of functional analysis and its many applications, such as Fourier analysis and the theory of partial differential equations. It follows that not only budding analysts have need of a clear understanding of the construction and properties of measures and integrals, but also that those who wish to contribute seriously to the applications of analytical methods in a wide variety of areas of mathematics, physics, electronics, engineering and, most recently, finance, need to study the underlying theory with some care. We have found remarkably few texts in the current literature which aim explicitly to provide for these needs, at a level accessible to current under- graduates. There are many good books on modern probability theory, and increasingly they recognize the need for a strong grounding in the tools we develop in this book, but all too often the treatment is either too advanced for an undergraduate audience or else somewhat perfunctory. We hope therefore that the current text will not be regarded as one which fills a much-needed gap in the literature! One fundamental decision in developing a treatment of integration is whether to begin with measures or integrals, i.e.
    [Show full text]
  • The Axiom of Choice and Its Implications
    THE AXIOM OF CHOICE AND ITS IMPLICATIONS KEVIN BARNUM Abstract. In this paper we will look at the Axiom of Choice and some of the various implications it has. These implications include a number of equivalent statements, and also some less accepted ideas. The proofs discussed will give us an idea of why the Axiom of Choice is so powerful, but also so controversial. Contents 1. Introduction 1 2. The Axiom of Choice and Its Equivalents 1 2.1. The Axiom of Choice and its Well-known Equivalents 1 2.2. Some Other Less Well-known Equivalents of the Axiom of Choice 3 3. Applications of the Axiom of Choice 5 3.1. Equivalence Between The Axiom of Choice and the Claim that Every Vector Space has a Basis 5 3.2. Some More Applications of the Axiom of Choice 6 4. Controversial Results 10 Acknowledgments 11 References 11 1. Introduction The Axiom of Choice states that for any family of nonempty disjoint sets, there exists a set that consists of exactly one element from each element of the family. It seems strange at first that such an innocuous sounding idea can be so powerful and controversial, but it certainly is both. To understand why, we will start by looking at some statements that are equivalent to the axiom of choice. Many of these equivalences are very useful, and we devote much time to one, namely, that every vector space has a basis. We go on from there to see a few more applications of the Axiom of Choice and its equivalents, and finish by looking at some of the reasons why the Axiom of Choice is so controversial.
    [Show full text]
  • (Measure Theory for Dummies) UWEE Technical Report Number UWEETR-2006-0008
    A Measure Theory Tutorial (Measure Theory for Dummies) Maya R. Gupta {gupta}@ee.washington.edu Dept of EE, University of Washington Seattle WA, 98195-2500 UWEE Technical Report Number UWEETR-2006-0008 May 2006 Department of Electrical Engineering University of Washington Box 352500 Seattle, Washington 98195-2500 PHN: (206) 543-2150 FAX: (206) 543-3842 URL: http://www.ee.washington.edu A Measure Theory Tutorial (Measure Theory for Dummies) Maya R. Gupta {gupta}@ee.washington.edu Dept of EE, University of Washington Seattle WA, 98195-2500 University of Washington, Dept. of EE, UWEETR-2006-0008 May 2006 Abstract This tutorial is an informal introduction to measure theory for people who are interested in reading papers that use measure theory. The tutorial assumes one has had at least a year of college-level calculus, some graduate level exposure to random processes, and familiarity with terms like “closed” and “open.” The focus is on the terms and ideas relevant to applied probability and information theory. There are no proofs and no exercises. Measure theory is a bit like grammar, many people communicate clearly without worrying about all the details, but the details do exist and for good reasons. There are a number of great texts that do measure theory justice. This is not one of them. Rather this is a hack way to get the basic ideas down so you can read through research papers and follow what’s going on. Hopefully, you’ll get curious and excited enough about the details to check out some of the references for a deeper understanding.
    [Show full text]
  • Measure Theory and Probability
    Measure theory and probability Alexander Grigoryan University of Bielefeld Lecture Notes, October 2007 - February 2008 Contents 1 Construction of measures 3 1.1Introductionandexamples........................... 3 1.2 σ-additive measures ............................... 5 1.3 An example of using probability theory . .................. 7 1.4Extensionofmeasurefromsemi-ringtoaring................ 8 1.5 Extension of measure to a σ-algebra...................... 11 1.5.1 σ-rings and σ-algebras......................... 11 1.5.2 Outermeasure............................. 13 1.5.3 Symmetric difference.......................... 14 1.5.4 Measurable sets . ............................ 16 1.6 σ-finitemeasures................................ 20 1.7Nullsets..................................... 23 1.8 Lebesgue measure in Rn ............................ 25 1.8.1 Productmeasure............................ 25 1.8.2 Construction of measure in Rn. .................... 26 1.9 Probability spaces ................................ 28 1.10 Independence . ................................. 29 2 Integration 38 2.1 Measurable functions.............................. 38 2.2Sequencesofmeasurablefunctions....................... 42 2.3 The Lebesgue integral for finitemeasures................... 47 2.3.1 Simplefunctions............................ 47 2.3.2 Positivemeasurablefunctions..................... 49 2.3.3 Integrablefunctions........................... 52 2.4Integrationoversubsets............................ 56 2.5 The Lebesgue integral for σ-finitemeasure.................
    [Show full text]
  • The Radon-Nikodym Theorem
    The Radon-Nikodym Theorem Yongheng Zhang Theorem 1 (Johann Radon-Otton Nikodym). Let (X; B; µ) be a σ-finite measure space and let ν be a measure defined on B such that ν µ. Then there is a unique nonnegative measurable function f up to sets of µ-measure zero such that Z ν(E) = fdµ, E for every E 2 B. f is called the Radon-Nikodym derivative of ν with respect to µ and it is often dν denoted by [ dµ ]. The assumption that the measure µ is σ-finite is crucial in the theorem (but we don't have this requirement directly for ν). Consider the following two examples in which the µ's are not σ-finite. Example 1. Let (R; M; ν) be the Lebesgue measure space. Let µ be the counting measure on M. So µ is not σ-finite. For any E 2 M, if µ(E) = 0, then E = ; and hence ν(E) = 0. This shows ν µ. Do we have the associated Radon-Nikodym derivative? Never. Suppose we have it and call it R R f, then for each x 2 R, 0 = ν(fxg) = fxg fdµ = fxg fχfxgdµ = f(x)µ(fxg) = f(x). Hence, f = 0. R This means for every E 2 M, ν(E) = E 0dµ = 0, which contradicts that ν is the Lebesgue measure. Example 2. ν is still the same measure above but we let µ to be defined by µ(;) = 0 and µ(A) = 1 if A 6= ;. Clearly, µ is not σ-finite and ν µ.
    [Show full text]
  • Delta Functions and Distributions
    When functions have no value(s): Delta functions and distributions Steven G. Johnson, MIT course 18.303 notes Created October 2010, updated March 8, 2017. Abstract x = 0. That is, one would like the function δ(x) = 0 for all x 6= 0, but with R δ(x)dx = 1 for any in- These notes give a brief introduction to the mo- tegration region that includes x = 0; this concept tivations, concepts, and properties of distributions, is called a “Dirac delta function” or simply a “delta which generalize the notion of functions f(x) to al- function.” δ(x) is usually the simplest right-hand- low derivatives of discontinuities, “delta” functions, side for which to solve differential equations, yielding and other nice things. This generalization is in- a Green’s function. It is also the simplest way to creasingly important the more you work with linear consider physical effects that are concentrated within PDEs, as we do in 18.303. For example, Green’s func- very small volumes or times, for which you don’t ac- tions are extremely cumbersome if one does not al- tually want to worry about the microscopic details low delta functions. Moreover, solving PDEs with in this volume—for example, think of the concepts of functions that are not classically differentiable is of a “point charge,” a “point mass,” a force plucking a great practical importance (e.g. a plucked string with string at “one point,” a “kick” that “suddenly” imparts a triangle shape is not twice differentiable, making some momentum to an object, and so on.
    [Show full text]
  • Harmonic Analysis Meets Geometric Measure Theory
    Geometry of Measures: Harmonic Analysis meets Geometric Measure Theory T. Toro ∗ 1 Introduction One of the central questions in Geometric Measure Theory is the extend to which the regularity of a measure determines the geometry of its support. This type of question was initially studied by Besicovitch, and then pursued by many authors among others Marstrand, Mattila and Preiss. These authors focused their attention on the extend to which the behavior of the density ratio of a m given Radon measure µ in R with respect to Hausdorff measure determines the regularity of the µ(B(x;r)) m support of the measure, i.e. what information does the quantity rs for x 2 R , r > 0 and s > 0 encode. In the context of the study of minimizers of area, perimeter and similar functionals the correct density ratio is monotone, the density exists and its behavior is the key to study the regularity and structure of these minimizers. This question has also been studied in the context of Complex Analysis. In this case, the Radon measure of interest is the harmonic measure. The general question is to what extend the structure of the boundary of a domain can be fully understood in terms of the behavior of the harmonic measure. Some of the first authors to study this question were Carleson, Jones, Wolff and Makarov. The goal of this paper is to present two very different type of problems, and the unifying techniques that lead to a resolution of both. In Section 2 we look at the history of the question initially studied by Besicovitch in dimension 2, as well as some of its offsprings.
    [Show full text]
  • Representation of the Dirac Delta Function in C(R)
    Representation of the Dirac delta function in C(R∞) in terms of infinite-dimensional Lebesgue measures Gogi Pantsulaia∗ and Givi Giorgadze I.Vekua Institute of Applied Mathematics, Tbilisi - 0143, Georgian Republic e-mail: [email protected] Georgian Technical University, Tbilisi - 0175, Georgian Republic g.giorgadze Abstract: A representation of the Dirac delta function in C(R∞) in terms of infinite-dimensional Lebesgue measures in R∞ is obtained and some it’s properties are studied in this paper. MSC 2010 subject classifications: Primary 28xx; Secondary 28C10. Keywords and phrases: The Dirac delta function, infinite-dimensional Lebesgue measure. 1. Introduction The Dirac delta function(δ-function) was introduced by Paul Dirac at the end of the 1920s in an effort to create the mathematical tools for the development of quantum field theory. He referred to it as an improper functional in Dirac (1930). Later, in 1947, Laurent Schwartz gave it a more rigorous mathematical definition as a spatial linear functional on the space of test functions D (the set of all real-valued infinitely differentiable functions with compact support). Since the delta function is not really a function in the classical sense, one should not consider the value of the delta function at x. Hence, the domain of the delta function is D and its value for f ∈ D is f(0). Khuri (2004) studied some interesting applications of the delta function in statistics. The purpose of the present paper is an introduction of a concept of the Dirac delta function in the class of all continuous functions defined in the infinite- dimensional topological vector space of all real valued sequences R∞ equipped arXiv:1605.02723v2 [math.CA] 14 May 2016 with Tychonoff topology and a representation of this functional in terms of infinite-dimensional Lebesgue measures in R∞.
    [Show full text]
  • A User-Friendly Introduction to Lebesgue Measure and Integration Gail S
    STUDENT MATHEMATICAL LIBRARY Volume 78 A User-Friendly Introduction to Lebesgue Measure and Integration Gail S. Nelson http://dx.doi.org/10.1090/stml/078 STUDENT MATHEMATICAL LIBRARY Volume 78 A User-Friendly Introduction to Lebesgue Measure and Integration Gail S. Nelson American Mathematical Society Providence, Rhode Island Editorial Board Satyan L. Devadoss John Stillwell (Chair) Erica Flapan Serge Tabachnikov 2010 Mathematics Subject Classification. Primary 26-XX, 28-XX. For additional information and updates on this book, visit www.ams.org/bookpages/stml-78 Library of Congress Cataloging-in-Publication Data Nelson, Gail Susan. A user-friendly introduction to Lebesgue measure and integration / Gail S. Nelson. pages cm. – (Student mathematical library ; volume 78) Includes bibliographical references and index. ISBN 978-1-4704-2199-1 (alk. paper) 1. Measure theory. 2. Lebesgue integral. 3. Integration, Functional. I. Title. QC20.7.M43N45 2015 515.83–dc23 2015022834 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Permissions to reuse portions of AMS publication content are handled by Copyright Clearance Center’s RightsLink service. For more information, please visit: http:// www.ams.org/rightslink. Send requests for translation rights and licensed reprints to reprint-permission @ams.org.
    [Show full text]
  • The Axiom of Choice
    THE AXIOM OF CHOICE THOMAS J. JECH State University of New York at Bufalo and The Institute for Advanced Study Princeton, New Jersey 1973 NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM LONDON AMERICAN ELSEVIER PUBLISHING COMPANY, INC. - NEW YORK 0 NORTH-HOLLAND PUBLISHING COMPANY - 1973 AN Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the Copyright owner. Library of Congress Catalog Card Number 73-15535 North-Holland ISBN for the series 0 7204 2200 0 for this volume 0 1204 2215 2 American Elsevier ISBN 0 444 10484 4 Published by: North-Holland Publishing Company - Amsterdam North-Holland Publishing Company, Ltd. - London Sole distributors for the U.S.A. and Canada: American Elsevier Publishing Company, Inc. 52 Vanderbilt Avenue New York, N.Y. 10017 PRINTED IN THE NETHERLANDS To my parents PREFACE The book was written in the long Buffalo winter of 1971-72. It is an attempt to show the place of the Axiom of Choice in contemporary mathe- matics. Most of the material covered in the book deals with independence and relative strength of various weaker versions and consequences of the Axiom of Choice. Also included are some other results that I found relevant to the subject. The selection of the topics and results is fairly comprehensive, nevertheless it is a selection and as such reflects the personal taste of the author. So does the treatment of the subject. The main tool used throughout the text is Cohen’s method of forcing.
    [Show full text]
  • 1. Σ-Algebras Definition 1. Let X Be Any Set and Let F Be a Collection Of
    1. σ-algebras Definition 1. Let X be any set and let F be a collection of subsets of X. We say that F is a σ-algebra (on X), if it satisfies the following. (1) X 2 F. (2) If A 2 F, then Ac 2 F. 1 (3) If A1;A2; · · · 2 F, a countable collection, then [n=1An 2 F. In the above situation, the elements of F are called measurable sets and X (or more precisely (X; F)) is called a measurable space. Example 1. For any set X, P(X), the set of all subsets of X is a σ-algebra and so is F = f;;Xg. Theorem 1. Let X be any set and let G be any collection of subsets of X. Then there exists a σ-algebra, containing G and smallest with respect to inclusion. Proof. Let S be the collection of all σ-algebras containing G. This set is non-empty, since P(X) 2 S. Then F = \A2SA can easily be checked to have all the properties asserted in the theorem. Remark 1. In the above situation, F is called the σ-algebra generated by G. Definition 2. Let X be a topological space (for example, a metric space) and let B be the σ-algebra generated by the set of all open sets of X. The elements of B are called Borel sets and (X; B), a Borel measurable space. Definition 3. Let (X; F) be a measurable space and let Y be any topo- logical space and let f : X ! Y be any function.
    [Show full text]
  • Chapter 6 Differentiation of Measures and Functions
    Chapter 6 Differentiation of Measures and Functions This chapter is concerned with the differentiation theory of Radon measures. In the first two sections we introduce the Radon measures and discuss two covering theorems, the main technical tools in differentiation theory. The Vitali's covering theorem applies to the Lebesgue measure, while the more sophisticated Besicovitch's covering theorem applies to Radon measures. In Section 3 we prove a version of Radon-Nikodym theorem for Radon measures. It differs from the version in Chapter 5 for now there is a good description of the Radon-Nikodym derivative. As application we deduce Lebsegue-Besicovitch differen- tiation theorem in Section 4. Next we study the differentiability properties of functions in R. First, we show how to relate increasing functions and Radon measures on the real line in Section 5. In Section 6 the concept of functions of bounded variations and absolutely continuous functions are recalled. The former is identified to be those integrable functions whose weak derivatives are signed Radon measures and the latter are those whose weak derivatives are integrable functions. 6.1 Radon Measures Recall that Riesz representation theorem asserts that for a positive linear functional Λ on the space Cc(X) where X is a locally compact Hausdorff space, there exists an Borel outer measure λ satisfying Z Λf = fdλ, for all f 2 Cc(X): We used to call λ the Riesz measure associated to Λ. Furthermore, λ enjoys the following regularity properties: n (a) λ(E) = inf λ(G): E ⊂ G; G open for every E ⊂ R . 1 2 CHAPTER 6.
    [Show full text]